For what combination of parallel and series resistors will resistors of 25 ohms, 100 ohms, 50 ohms, and 50 ohms have a total resistance of 62.5 ohms?
To determine the combination of parallel and series resistors that will result in a total resistance of 62.5 ohms using the given resistors (25 ohms, 100 ohms, 50 ohms, and 50 ohms), we need to apply the concepts of parallel and series resistor combinations.
In a series combination, resistors are arranged sequentially, one after the other, like a chain. The total resistance of a series combination is equal to the sum of the individual resistances. So, for a series combination, the total resistance is given by:
Total Resistance (in series) = R1 + R2 + R3 + ...
On the other hand, in a parallel combination, resistors share the same two endpoints, and the total resistance is determined by the reciprocal of the sum of the reciprocals of the individual resistances. Mathematically, for a parallel combination, the total resistance is given by:
Total Resistance (in parallel) = 1 / (1/R1 + 1/R2 + 1/R3 + ...)
Now, let's find the combination of parallel and series resistors that will give us a total resistance of 62.5 ohms.
First, we need to identify how we can group the given resistors to achieve the desired total resistance. We can try different combinations and calculate the resulting total resistance until it matches 62.5 ohms.
Let's try a series combination with the resistors of 25 ohms, 100 ohms, and 50 ohms (50 ohms omitted temporarily). The total resistance would be:
Total Resistance (Series Combination) = 25 ohms + 100 ohms + 50 ohms = 175 ohms
However, this value is higher than the desired 62.5 ohms. We can conclude that using these resistors only in a series combination will not yield the desired total resistance.
Now, let's consider a parallel combination using these resistors: 25 ohms, 100 ohms, and 50 ohms. The total resistance would be:
Total Resistance (Parallel Combination) = 1 / (1/25 ohms + 1/100 ohms + 1/50 ohms)
By calculating this expression, we find that the total resistance would be approximately 14.29 ohms. Again, this value is not equal to 62.5 ohms.
Now, let's try a combination of series and parallel resistors. We can use the 25 ohm and 100 ohm resistors in series, and combine them in parallel with the 50 ohm resistor. In this configuration, the total resistance would be:
Total Resistance = (25 ohms + 100 ohms) || 50 ohms
|| denotes a parallel combination.
Now, we can calculate the total resistance:
Total Resistance = (125 ohms) || 50 ohms
To calculate the parallel combination, we need to find the reciprocal of the sum of the reciprocals:
Total Resistance = 1 / (1/125 ohms + 1/50 ohms)
Total Resistance ≈ 1 / (0.008 + 0.02)
Total Resistance ≈ 1 / 0.028
Total Resistance ≈ 35.71 ohms
Unfortunately, in this configuration, the total resistance is not equal to 62.5 ohms either.
After trying various combinations, we can conclude that there is no simple combination of the given resistors (25 ohms, 100 ohms, 50 ohms, and 50 ohms) in series or parallel that will result in a total resistance of exactly 62.5 ohms.
However, keep in mind that more complex resistor networks using additional resistors or using different values for the given resistors may allow you to achieve the desired total resistance.